Is there a natural way to define the terminal state from the MDP transition probabilities $p(s',r|s,a)$?
I don't know if there is a general definition of the terminal state based on the MDP transition probabilities.
But remember that we define our MDP problem in a $\mathbb{S}$ set of all possible states and a $\mathbb{A}(s)$ representing the set of all possible actions for each state. Based on that, probably there aren't any possible actions for the terminal state, so the transition probability $p(s', r |s_T, a)$ can be undefined. Based on that, in episodic tasks we need to define a the set of all possible terminal states $\mathbb{S}^+$ and distinguish this with $\mathbb{S}$.